Part 1 Lecture Notes on Geometric Crysta

These lecture notes present an in-depth exploration of geometric crystals and their combinatorial analogues, primarily focusing on the birational actions of the Weyl group on affine algebraic varieties. The objective of the work is to tackle the problem of constructing a birational action of the Weyl group on various function spaces, emphasizing the importance of the interactions between algebraic tori and crystal bases. This is achieved through a methodological approach that leverages the authors' prior research over the years while introducing conjectures that connect local Langlands conjectures with algebro-geometric distributions. The results demonstrate that, for each algebraic representation of the Langlands dual group, a W-equivariant algebro-geometric distribution can indeed be constructed, which highlights its implications for the resolution of outstanding conjectures within the field. These findings advance the understanding of geometric representations and provide significant contributions to the area of combinatorial algebra.